Perhaps you are referring to modern nonstandard analysis, which I am sure has equivalent definition of limits. I will note that in standard calculus Epsilon-Delta is not only used in defining derivatives, but also used in defining function limits, which of course means used in defining integrals.
Just out of curiosity, in nonstandard analysis is there an equivalent of Lebesgue integral?
Just out of curiosity, in nonstandard analysis is there an equivalent of Lebesgue integral?